Quandaries and Queries


Can you help me understand why the area of a equilateral triangle is the square root of 3 divided by 4 times the lenght of the side squared?





I drew a diagram of an equilateral triangle.

Draw a line from the vertex to the midpoint of the base.

Cut along this line and move half of the triangle to form a rectangle.

If a is the length of a side of the equilateral triangle and h the height then this rectangle has area

 1/2  a  h

The triangle below, which is half of the rectangle, is a right triangle,

and hence, by Pythagoras' theorem

a2 = h2 + ( a/2)2

Thus h = sqrt( 3/4 a2) =  sqrt(3)/2 a. Hence the area of the triangle is

 1/2  a  h = sqrt(3)/4 a2



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